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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.69

Online Matroid Embeddings

Authors: Andrés Cristi, Paul Dütting, Robert Kleinberg, Renato Paes Leme, and Neel Patel

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. The existence of such embedding enables a reduction from the version of the matroid secretary problem where the matroid is unknown to the version where the matroid is known in advance. We establish the existence of such an embedding for binary matroids, and use it to relate variants of the binary matroid secretary problem to each other, showing that seemingly simpler problems are in fact equivalent to seemingly harder ones (up to constant-factors). Specifically, we show this to be the case for the version of the matroid secretary problem in which the binary matroid is not known in advance, and where it is known in advance. We also show that the version with known matroid structure is equivalent to the problem where weights are not fully adversarial but drawn from a known pairwise-independent distribution.

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Andrés Cristi, Paul Dütting, Robert Kleinberg, Renato Paes Leme, and Neel Patel. Online Matroid Embeddings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 69:1-69:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cristi_et_al:LIPIcs.ICALP.2026.69,
  author =	{Cristi, Andr\'{e}s and D\"{u}tting, Paul and Kleinberg, Robert and Paes Leme, Renato and Patel, Neel},
  title =	{{Online Matroid Embeddings}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{69:1--69:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.69},
  URN =		{urn:nbn:de:0030-drops-264585},
  doi =		{10.4230/LIPIcs.ICALP.2026.69},
  annote =	{Keywords: Matroids, Secretary Problem, Online Algorithm}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.51

On Sparsification of Stochastic Packing Problems

Authors: Shaddin Dughmi, Yusuf Hakan Kalayci, and Neel Patel

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

Motivated by recent progress on stochastic matching with few queries, we embark on a systematic study of the sparsification of stochastic packing problems more generally. Specifically, we consider packing problems where elements are independently active with a given probability p, and ask whether one can (non-adaptively) compute a "sparse" set of elements guaranteed to contain an approximately optimal solution to the realized (active) subproblem. We seek structural and algorithmic results of broad applicability to such problems. Our focus is on computing sparse sets containing on the order of d feasible solutions to the packing problem, where d is linear or at most polynomial in 1/p. Crucially, we require d to be independent of the number of elements, or any parameter related to the "size" of the packing problem. We refer to d as the "degree" of the sparsifier, as is consistent with graph theoretic degree in the special case of matching. First, we exhibit a generic sparsifier of degree 1/p based on contention resolution. This sparsifier’s approximation ratio matches the best contention resolution scheme (CRS) for any packing problem for additive objectives, and approximately matches the best monotone CRS for submodular objectives. Second, we embark on outperforming this generic sparsifier for additive optimization over matroids and their intersections, as well as weighted matching. These improved sparsifiers feature different algorithmic and analytic approaches, and have degree linear in 1/p. In the case of a single matroid, our sparsifier tends to the optimal solution. In the case of weighted matching, we combine our contention-resolution-based sparsifier with technical approaches of prior work to improve the state of the art ratio from 0.501 to 0.536. Third, we examine packing problems with submodular objectives. We show that even the simplest such problems do not admit sparsifiers approaching optimality. We then outperform our generic sparsifier for some special cases with submodular objectives.

Cite as

Shaddin Dughmi, Yusuf Hakan Kalayci, and Neel Patel. On Sparsification of Stochastic Packing Problems. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dughmi_et_al:LIPIcs.ICALP.2023.51,
  author =	{Dughmi, Shaddin and Kalayci, Yusuf Hakan and Patel, Neel},
  title =	{{On Sparsification of Stochastic Packing Problems}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{51:1--51:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.51},
  URN =		{urn:nbn:de:0030-drops-181036},
  doi =		{10.4230/LIPIcs.ICALP.2023.51},
  annote =	{Keywords: Stochastic packing, sparsification, matroid}
}

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Online Matroid Embeddings

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On Sparsification of Stochastic Packing Problems

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